The Unapologetic Mathematician

Mathematics for the interested outsider

Adjoint Transformations

Since an inner product on a finite-dimensional vector space V is a bilinear form, it provides two isomorphisms from V to its dual V^*. And since an inner product is a symmetric bilinear form, these two isomorphisms are identical. But since duality is a (contravariant) functor, we have a dual transformation T^*:W^*\rightarrow V^* for every linear transformation T:V\rightarrow W. So what happens when we put these two together?

Say we start with linear transformation T:V\rightarrow W. We’ll build up a transformation from W to V which we’ll call the “adjoint” to T. First we have the isomorphism W\rightarrow W^*. Then we follow this with the dual transformation T^*:W^*\rightarrow V^*. Finally, we use the isomorphism V^*\rightarrow V. We’ll write T^* for this composite, and rely on context to tell us whether we mean the dual or the adjoint (but because of the isomorphisms they’re secretly the same thing).

So why is this the adjoint? Let’s say we have vectors v\in V and w\in W. Then it turns out that

\displaystyle\left\langle T(v),w\right\rangle_W=\left\langle v,T^*(w)\right\rangle_V

which should recall the relation between two adjoint functors. An important difference here is that there is no distinction between left- and right-adjoint transformations. The adjoint of an adjoint is the original transformation back again: \left(T^*\right)^*=T. This follows if we use the symmetry of the inner products on the relation above

\displaystyle\left\langle w,T(v)\right\rangle_W=\left\langle T^*(w),v\right\rangle_V=\left\langle w,T^{**}(v)\right\rangle_W

Then since \left\langle w,\left[T-T^{**}\right](v)\right\rangle_W=0 and the inner product on W is nondegenerate, we must have T-T^{**} sending every v to the zero vector in W. Thus T=T^{**}.

So let’s show this adjoint condition in the first place. On the left side, we have the result of applying the linear functional \langle\underline{\hphantom{X}},w\rangle_W\in W^* to the vector T(V). But this linear functional is simply the image of the vector w under the isomorphism W\rightarrow W^*. So on the left, we’ve calculated the result of first applying T to v, and then applying this linear functional.

But the way we defined the dual transformation was such that we can instead apply the dual T^* to the linear functional \langle\underline{\hphantom{X}},w\rangle_W, and then apply the resulting functional to v, and we’ll get the same result. And the isomorphism V^*\rightarrow V tells us that there is some vector in v_1\in V so that the linear functional we’re now applying to v is \langle\underline{\hphantom{X}},v_1\rangle_V. That is, our value will be \langle v,v_1\rangle_V for some vector v_1. Which one? The one we defined as T^*(w).

I’ll admit that it sometimes takes a little getting used to the way adjoints and duals are the same, and also the subtleties of how they’re distinct. But it sinks in soon enough.

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May 22, 2009 - Posted by | Algebra, Linear Algebra

7 Comments »

  1. [...] of Adjoints Many of the properties of the adjoint construction follow immediately from the contravariant functoriality of the duality we used in its [...]

    Pingback by Properties of Adjoints « The Unapologetic Mathematician | May 25, 2009 | Reply

  2. [...] We left off defining what we mean by a matrix element of a linear transformation. Let’s see how this relates to adjoints. [...]

    Pingback by The Matrix of the Adjoint « The Unapologetic Mathematician | June 22, 2009 | Reply

  3. [...] Let’s now consider a single inner-product space and a linear transformation . Its adjoint is another linear transformation . This opens up the possibility that might be the same [...]

    Pingback by Self-Adjoint Transformations « The Unapologetic Mathematician | June 23, 2009 | Reply

  4. [...] this is an inner product, and so we have the adjoint property, allowing us to rewrite it as . This is the pairing of the bra and the ket . But this latter ket [...]

    Pingback by Dirac Notation III « The Unapologetic Mathematician | July 6, 2009 | Reply

  5. [...] to the basis of . And this is actually just as we expect, since the transpose is actually the adjoint transformation, which automatically connects the dual [...]

    Pingback by Cotangent Vectors, Differentials, and the Cotangent Bundle « The Unapologetic Mathematician | April 13, 2011 | Reply

  6. [...] one. Let’s say that is an inner product on a vector space . As we mentioned when discussing adjoint transformations, this gives us an isomorphism from to its dual space . That is, when we have a metric floating [...]

    Pingback by Inner Products on 1-Forms « The Unapologetic Mathematician | October 1, 2011 | Reply

  7. [...] the other hand, the codifferential is (sort of) the adjoint to the differential. Adjointness would mean that if is a -form and is a -form, [...]

    Pingback by The Codifferential « The Unapologetic Mathematician | October 21, 2011 | Reply


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